What “Balanced on Average” Means
Random assignment is valuable because the assignment mechanism does not select people for treatment according to their pretreatment characteristics. If the mechanism is properly implemented, a participant’s age, disease severity, or other baseline feature does not make that participant more likely to be assigned to one arm rather than the other. This gives randomized groups a useful property even when researchers do not know every characteristic that could affect the outcome.
The phrase “on average” needs care. It refers to the average over all assignments that could have been generated by the study’s randomization mechanism, holding the enrolled participants and their pretreatment characteristics fixed. It does not say that the two groups in the one assignment actually observed must have identical characteristics.
For the derivation, regard the \(N\) enrolled units and their baseline values \(X_1,\ldots,X_N\) as fixed. Let \(Z_i=1\) if unit \(i\) is assigned to treatment and \(Z_i=0\) otherwise. Under complete randomization, each unit has probability \(n_1/N\) of assignment to treatment. The assignment indicators are not mutually independent: because exactly \(n_1\) units must be treated, assigning one person to treatment slightly constrains the assignments available to the others.
Why Group Means Match in Expectation
The full study’s mean baseline characteristic is \(\bar X=N^{-1}\sum_{i=1}^{N}X_i\). The treatment-group mean depends on the assignment: \(\bar X_1(Z)=n_1^{-1}\sum_{i=1}^{N}Z_iX_i\). Since \(X_i\) is fixed and \(E_R(Z_i)=n_1/N\), where \(E_R\) means expectation over the randomization mechanism, linearity of expectation gives:
The control group contains \(N-n_1\) units. Each unit has probability \((N-n_1)/N\) of being assigned to control. Applying the same argument to its mean, \(\bar X_0(Z)\), gives \(E_R\{\bar X_0(Z)\}=\bar X\). Consequently:
This is a statement about the expected difference, not a claim that each possible assignment has a zero difference. It applies to any numerical pretreatment characteristic. It also applies to a categorical characteristic by using an indicator: for example, set \(X_i=1\) for a participant with a specified condition and \(X_i=0\) otherwise. The group mean of that indicator is the group’s proportion with the condition, so the expected proportion in either group equals the study-wide proportion.
The same reasoning applies to any fixed function of baseline information, such as a risk score \(g(X_i)\), because it is simply another fixed value for each unit. In potential-outcomes language, assignment is generated without selecting units according to their potential outcomes \(Y_i(0)\) and \(Y_i(1)\), which are defined before assignment. Thus their group averages are balanced in expectation as well. This does not require the researcher to have measured every prognostic characteristic.
Worked Example: Expected Age Difference in a Six-Person Trial
Suppose six hypothetical participants have baseline ages of 20, 22, 24, 26, 28, and 30 years. Three are assigned to treatment by complete randomization. Find the expected treatment-group mean age and expected treatment-minus-control difference.
The study-wide mean age is \((20+22+24+26+28+30)/6=150/6=25\) years. Each participant has probability \(3/6=0.5\) of treatment assignment. The expected sum of the ages assigned to treatment is therefore:
There are exactly three treated participants, so the expected treatment-group mean is \(75/3=25\) years. The control-group mean also has expectation 25 years, and the expected difference is \(25-25=0\) years. This does not imply that an actual allocation must have equal means. For example, assigning ages 20, 22, and 24 to treatment gives a treatment mean of 22 years and a control mean of 28 years: a difference of \(-6\) years. That allocation is compatible with randomization despite its unequal means.
Balance Applies to More Than Means
The result is not limited to a continuous measurement such as age. For a binary characteristic, the group mean of its zero-one indicator is a proportion. For a categorical variable with several levels, use one indicator per level. Each level’s expected group proportion equals its study-wide proportion. Random assignment also balances fixed transformations of baseline values in expectation, although it does not guarantee that every possible summary will be exactly equal in a realized allocation.
Worked Example: A Binary Pretreatment Characteristic
In a hypothetical trial of 12 people, eight have a baseline risk factor and four do not. Complete randomization assigns six people to treatment and six to control. Find the expected number and proportion with the risk factor in each arm. Then assess one possible realized allocation with five risk-factor participants in treatment.
Let \(X_i=1\) for a participant with the risk factor and \(X_i=0\) otherwise. The study-wide proportion is \(8/12=0.667\). Each person has probability \(6/12=0.5\) of treatment assignment, so the expected treated count with the risk factor is \(8(0.5)=4\). The expected treatment proportion is \(4/6=0.667\). By the same calculation, the expected control count is 4 and the expected control proportion is \(4/6=0.667\).
Now suppose the realized treatment group contains five of the eight risk-factor participants. The other three are in control. The realized proportions are \(5/6=0.833\) in treatment and \(3/6=0.500\) in control, for a difference of \(0.833-0.500=0.333\), or 33.3 percentage points. This allocation does not contradict the expectation result: that result averages over the possible assignments, not just this one.
Worked Example: A Prognostic Baseline Score
Consider four hypothetical participants with fixed baseline risk scores \(X_i=1,2,5,8\) points and fixed untreated potential outcomes \(Y_i(0)=4,6,12,18\) outcome units. Two participants are assigned to treatment by complete randomization. Calculate the expected treatment and control means of both the risk score and the untreated potential outcome.
Each participant has probability \(2/4=0.5\) of treatment assignment. The study-wide risk-score mean is \((1+2+5+8)/4=4\) points. Therefore the expected treatment-group risk-score sum is \(0.5(1+2+5+8)=8\) points, and dividing by two gives an expected mean of 4 points. The control-group expected mean is also 4 points.
The study-wide mean untreated potential outcome is \((4+6+12+18)/4=40/4=10\) outcome units. The expected sum of untreated potential outcomes among the two treated-assigned participants is \(0.5(4+6+12+18)=20\) units. Dividing by two gives an expected mean of 10 units; the control-assigned group has the same expected mean. Randomization therefore balances these fixed prognostic attributes in expectation, even though one realized allocation need not balance either one exactly. The potential outcomes are used here to describe what is fixed before assignment; the calculation does not claim that both potential outcomes are observed for each person.
Conditions, Scope, and Comparisons
The derivation depends on the assignment mechanism, not on a favorable outcome model. Its conditions are straightforward but important:
- The treatment allocation is genuinely generated by the stated random mechanism, with each unit’s assignment probability as specified.
- The characteristics being called pretreatment are determined before assignment. A feature changed by treatment is not a baseline characteristic.
- The analyzed groups correspond to the randomized units and assignments. Exclusions or missing measurements related to assignment or baseline characteristics can make the analyzed comparison differ from the randomized comparison.
Random assignment is not the same as random sampling. Assignment determines which enrolled units receive treatment; sampling determines which units enter the study. A trial can balance its enrolled participants in expectation while still enrolling a group that differs from the wider population. Therefore, randomization alone does not establish that results generalize to people who were not enrolled.
| Feature | Random assignment | Nonrandom treatment assignment |
|---|---|---|
| How treatment is allocated | A specified chance mechanism determines assignment. | Assignment may depend on measured or unmeasured characteristics. |
| Baseline balance | Group summaries have the same expectation under the assignment mechanism. | No general assignment-based balance guarantee. |
| What the claim guarantees | No systematic sorting by baseline characteristics over possible assignments. | Balance requires additional design or analysis assumptions. |
| What it does not guarantee | Exact equality of groups in the one realized trial, or representativeness of the enrolled sample. | — |
This comparison concerns assignment, not every other feature of a study. Randomization protects against systematic treatment selection through baseline characteristics, but it cannot repair later departures from the assigned protocol or make incomplete measurements harmless. Nor does a chance mechanism require treatment and control indicators to be independent across people under fixed-size complete randomization.
A Common Mistake and a Practical Interpretation
Wrong reasoning: “Treatment was randomized, so the treatment and control groups must have the same baseline characteristics.” This confuses equality in expectation with equality in a single realized allocation. Randomization makes imbalances unsystematic with respect to the assignment mechanism; it does not rule them out.
Correction: State the claim at the right level: across the assignments permitted by the mechanism, each group’s expected baseline mean (or proportion) equals the study-wide value. Then describe the actual allocation as one realization. A difference in the realized baseline summaries is not, by itself, evidence that random assignment failed; it may occur under a correctly implemented chance mechanism.
The central benefit is that the assignment rule does not deliberately favor one arm with participants who have particular baseline causes of the outcome. This is a design-based reason for treating randomized comparisons differently from comparisons in which treatment selection may depend on such causes. It is not a guarantee that all observed groups look alike, and it does not remove the need to examine how the trial was carried out.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A complete-randomization trial assigns 10 of 20 participants to treatment. What is each participant’s probability of treatment assignment?
- A binary baseline feature occurs in 30 of 60 enrolled participants. What are the expected treatment and control proportions with the feature if 30 participants are assigned to each arm?
- Does the expectation result imply that the two groups in the observed trial must have identical mean ages? Explain.
- Why is random assignment not a guarantee that the enrolled sample represents the wider population?
- Under fixed-size complete randomization, are the assignment indicators for different participants mutually independent? Explain briefly.
Solutions. (1) The probability is \(10/20=0.5\) for each participant. (2) The study-wide proportion is \(30/60=0.5\), so each arm’s expected proportion is 0.5. (3) No. The equality is over possible assignments; one realized allocation can have different means. (4) Assignment determines treatment among enrolled participants, whereas sampling determines who enrolls. The assignment mechanism alone makes no claim about the enrollment process. (5) No. Because exactly 10 participants must be assigned to treatment, assigning one participant to treatment changes the remaining number of treatment places; the indicators are constrained by the fixed total.